EinsteinHilbertAction

Let $(\mathcal{M}, g)$ be a 4-dimensional, smooth, Lorentzian manifold $\mathcal{M}$ with an associated metric tensor $g_{\mu\nu}(x)$, a function of the coordinates $x$ on $\mathcal{M}$. Then the Einstein-Hilbert action $S$ is given by the integral

$$ S[g] = \frac{ \int_{\mathcal{M}} ( R(x) + \mathcal{L}_m(x) ) \sqrt{-\det(g(x))} d^4x}{2\kappa} $$

where:

1. $\kappa = 8\pi G$, with $G$ being Newton's gravitational constant.
2. $R = R(x)$ is the scalar curvature at point $x$, a real-valued function on $\mathcal{M}$ obtained from contracting the Ricci curvature tensor $R_{\mu\nu}(x)$, which itself is a contraction of the Riemann curvature tensor $R^\rho_{\sigma\mu\nu}(x)$.
3. $\sqrt{-\det(g)}$ is the square root of the negative of the determinant of the metric tensor $g_{\mu\nu}(x)$, ensuring the correct volume element in the integration over the manifold.
4. $d^4x$ represents the integration over the 4-dimensional manifold $\mathcal{M}$.
5. $\mathcal{L}_m$ is the Lagrangian density for any matter fields present, which contributes to the overall action but is not part of the Einstein-Hilbert term itself.